Causing a stir

Today’s homepage on the BBC’s website questions whether breast cancer screening does more harm than good (see here for the article). However, the article refers to sets of data which can provide misleading information if not dealt with carefully. When referring to people’s health, this can be very dangerous.

Consider the example below. The numbers are rounded, to make the maths easier, but are close – and more than adequately demonstrate the effect.

For any woman, the probability of having breast cancer is 1 per cent.

Of those who have cancer, screening will give a positive test result 90% of the time (10% of tests are incorrectly identified as negative).

Of those who do not have cancer, screening will still give a positive (a ‘false positive’) test result 9% of the time (91% of these tests correctly identify that cancer is not present).

So, with my maths teacher hat on, I ask the question “if a woman tests positive to a breast cancer screening, what is the (approximate) probability she has cancer?”

  1. 90%
  2. 80%
  3. 10%
  4. 1%

Take a moment to think through your answer…

[if this were a maths lesson, I would leave the question at this point for a while, to give the students an opportunity to figure out the answer; meanwhile, I might suggest that most gynaecologists would intuitively suggest either A or B, ref here]

To answer the question, it can be treated in terms of natural frequencies.

The article on the BBC refers to 2000 women, so I have referred to the same number of people here.

We ‘know’ that 1% of them have cancer, i.e. 20 people. Of those, we would expect 90% (18 people) to test positive when screened.

Of the 1980 people who don’t have cancer, we would still expect 178 (9% of 1980) to test positive when screened (these are the ‘false positives’).

In total, this means that 196 (18 + 178 = 196) women test positive, 18 of whom have cancer.

And 2 cancer sufferers would go undetected.

So the answer to the question is about 10%. 18 out of 196 is 9.18% (to 3 significant figures).

The point I am making here is that it is very easy to get lost in amongst all of the numbers. The media are very good at whipping this up and scaring the reader. But, when talking about people’s health, it is of paramount importance to be as clear and honest as possible.

It’s not like this is even new news. The BBC did the maths 30 months ago (see here), albeit with simpler numbers than used in this post. But after all, what use is an article whose headline is “Breast cancer screening under review” other than to generate a bit of scaremongering.

For information, the maths of conditional probabilities involved here is Bayes’ Theorem

Update 30th October 2012: The New York Times published this article, echoing the confusion, on 29th October 2012.

Being nice prevents tooth decay

An article in today’s Daily Telegraph makes reference to a study that was carried out in order to identify a link between violence in teenagers and their drinking of fizzy drinks.

Almost 2,000 14 to 18-year-olds from 20 schools in Boston were asked how many cans of non-diet fizzy drink they had consumed over the past week and if they had been violent towards a peer, sibling or partner over the previous year.

From this sample and these questions, the article concludes that fizzy drinks addicts* were more likely to be violent to their peers [58 per cent for addicts versus 35 per cent for non-addicts], siblings [43% vs 25%] or partners [27% vs 15%]. As such, the newspaper opted for the headline ‘Fizzy drinks make teenagers violent’. However, there is no evidence that this is the case; the headline could equally have been ‘Violence makes teenagers consume fizzy drinks’, although my preference would be the one used as the title for this entry.

I understand that newspapers need to generate money, and catchy headlines are a significant way of aiding this, but I become frustrated by how misleading they can be. Is it any wonder that students respond like this in their exam**, when the world is full of so much conflicting information?

2011-10-22_10

Thanks to David Spiegelhalter for bringing this article to my attention.

 

* An addict was classed as someone drinking more than fourteen fizzy drinks per week (two per day), whereas a non-addict as someone drinking a maximum of one can a week [quite what those in the middle are classed as, I am not sure].

** For information, the student actually achieved a grade C in their exam.

Bog maths

There are many ways to combine running with teaching maths, but frequency of portaloos is not one I have previously encountered (see here) – the opportunities are almost endless; ratio, formulae and calculus to name a few.

In fact, this could even form the basis of an entire project.

And if it all else fails, the paper could even be …  [yep, you know where this one is going]

What we can learn from Mo Farah

I have previously made reference to this quote, by Lance Armstrong:

Pain is temporary. It may last a minute, or an hour, or a day, or a year, but eventually it will subside and something else will take its place. If I quit, however, it lasts forever.

I believe that there are not many people as well qualified as Lance Armstrong to make such statements, but it was obviously visible at the 2011 Athletics World Championships in Daegu. Mo Farah has been a good athlete for about a decade. He has taken a long time to make the step up to become the UK’s number one distance runner and subsequently progressed further and demonstrated that he can compete against some of the best.

But for all of the personal goals which athletics allows you to achieve, running fast times and winning championship races are two very different practices. Mo Farah went into the 10 000m last Sunday as the fastest UK distance runner in history, as well as having beaten all of his obvious close competitors in the earlier parts of the year. The race went predominantly according to plan for 23½ of the 25 laps and, with 600m to go, he accelerated exactly as he had done throughout the season with much success. This video shows the outcome.

He was tracked by a runner whom he hadn’t competed against previously and was beaten in the final 100m.

With tired legs, it would have been easy to go home with the silver medal, and prepare for next year’s Olympics. But Mo Farah demonstrated that he is not a quitter by trying to rectify the situation at the earliest possible opportunity – and fitness allowed him to run on Wednesday and successfully qualify for the 5000m final the following Sunday.

Again, he ran the race sensibly for the first 11 of the 12½ laps. This video shows the outcome.

Contrast the difference in the two final laps. The latter is a demonstration of the discipline, mental strength and intelligence required to successfully compete at such a level; it almost makes the first appear naive and immature. He made a mistake, learnt from it, put it into practice and reaped the benefits.

So, as we start a new academic year, what better message can be sent to students? Work hard, learn from your mistakes, and you will be rewarded. And as for teachers, we can follow the same advice. Good luck to all.

NB. Mo Farah gets my vote for BBC Sports Personality of the year.

NB2. This could also form the basis of another (teaching) blog about the necessity and quality of rewards, but that will have to wait for another day.

NB3. For those who want to see the race in more detail, check http://athletics.channel4.com/

Tracking and feedback

As a teacher, I have always tried to maintain an electronic register. In the past I have used this to help me track students’ progress and write reports. But I don’t always use this information as well as I could and, as such, I sometimes end up with more information than the students themselves – when they are the ones who need to know how to improve.

To overcome this problem, I have recently cleaned up the ‘front end’, so that students (and/or their parents) could potentially access this information. The idea is that students would go to a web address, type in their unique ‘user information’ and they would have access to their current performance, targets and what they need to do in order to improve.

I would be grateful if any readers (teachers, parents, others) could take the time to have a look at the Excel file (here) and let me have their thoughts. Although the file currently only contains fictitious data, I have hidden the ‘back end’ in an attempt to prevent potential hackers (i.e. parents and students) from maliciously (or otherwise) seeing other people’s information in a complete version; I would therefore also be interested to know if anyone can identify any flaws in the file’s security.

While I appreciate this won’t necessarily have a direct impact on what goes on in the classroom, I think it could potentially improve my overall practice if used effectively. An apparent flaw could be the amount of automation in the process – and the potential lack of my input. However, the file has been set up so that I can enter as much (or as little) information as I choose, increasing in accuracy with more data entered. It also started as a completely blank ‘canvas’ so is specific to my own current requirements.

In case there are any problems with the file, I have uploaded some pictures of what the document currently looks like.

Thank you.

Elections

The recent referendum on AV, which caused much debate, provides many mathematical opportunities, and this video shows what happens in the event of a tie.

Personally, I would like to see a tie decided by rock-paper-scissors. This version claims to learn your behaviour and subsequently beat you – note that it is more likely to beat you if you play for a long time (whatever the definition of ‘long’ is). The image below shows how to defeat a human opponent, and maybe this should be shown to both candidates before playing.

Rock-paper-scissors-575x1310

So, how about this for a potential framework for a lesson on experimental probability:

(i) estimate the experimental probability of winning a game against a human
(ii) estimate the experimental probability of winning a game against a human when one player has seen a ‘strategy’ (i.e. has seen the image)
(iii) estimate the experimental probability of winning a game against a human when both players have seen the same ‘strategy’(or indeed a different strategy)
(iv) estimate the experimental probability of winning a game against a computer

Triangular chords

Looking forward to working on triangles this week with the help of James Blunt. Where else will you hear the lyrics “it must be those angles, put a smile on your face, not to mention the hypotenuse”.

I just hope this doesn’t encourage the misspelling of angles and angels.

Check the video here for info.

Beast Beyond Belief

One student in class today was factorising quadratic equations and was struggling, so I showed him the ‘Lizzie’ method http://tinyurl.com/2u6funv. He is one of the cleverer students and appreciates that there are often different ways to solve problems. He could also see that I was largely working it out as I was going along (well, I did tell him that this was the case), but as we approached the answer, he proclaimed:

“sir, you are a beast beyond belief”

I smiled to myself. I had managed to ‘do it’ and help him but, also, I had received praise from a student. It doesn’t happen very often, but reminds me of my belief that it is not the nature of the praise that is important – more the fact that there is an acknowledgement of ‘doing good’.

Many argue that some students don’t appreciate praise. However, I believe that while it may not be cool to receive praise, particularly in front of peers, the recognition (and confirmation) is hugely appreciated – even if only in secret. I also don’t believe the ‘size’ of the praise matters. While a monetary reward acts as a carrot to pull, in the same way that a prison sentence acts as a stick to push, a simple ‘merit’ or ‘demerit’ is enough to provide confirmation of good or bad. As such, my lessons tend to be full of lots of relatively meaningless (but well intentioned) positives (credits/good comments/commendations/merits etc.) and negatives (verbal and visual warnings).

My success rate is determined by my consistency of application.

This is war

Memo

I always find the first term (in the run up to Christmas) the most exhausting. And this seems to be exaggerated this year being at a new school. However, it is the small things that help keep me on the straight and narrow.

Like the Year 8 student who was ashamed to admit to his friends that he had a Facebook account, or the girl who wanted to start a religion called “Anya-ism” (pronounced aneurism), or the memo from the History department warning me amount the weapons my form will be bringing to school on Monday.
I love my job.